Volume 215, number 3
PHYSICS LETTERS B
22 December 1988
B O U N D STATE APPROACH FOR STRANGE DIBARYONS IN THE SKYRME MODEL J. K U N Z and P.J. M U L D E R S National Institute for Nuclear Physics and High Energy Physics, N1KHEF-K, P.O. Box 41882, NL-I O09 DB Amsterdam. The Netherlands
Received 26 July 1988; revised manuscript received 28 September 1988
The bound state approach to strange dibaryons in the Skyrme model is extended to baryon number n> 1. Kaon bound states are obtained in a (variational) axially symmetric SU (2) skyrmion background field. Collective quantization of isospin and spatial zero modes leads to dibaryon quantum states. These are classified in flavor multiplets.
The SU (2) Skyrme model works remarkably well in reproducing the properties of the nonstrange baryons [1,2]. Its straightforward extension to SU (3), where the classical S U ( 2 ) solution is embedded in the larger group and all SU (3) coordinates are treated as collective modes, fails in reproducing the SU (3) baryon spectrum [3 ]. Callan and Klebanov ( C K ) [4,5 ] showed that a phenomenologically good description of the baryon flavor SU ( 3 ) multiplets can be obtained in the Skyrme model when the strange pseudoscalar mesons, the kaons, are treated differently from the pions. This different treatment finds its motivation in the strong breaking o f the SU (3) flavor symmetry, since the kaons are much heavier than the pions. Therefore, only the SU (2) isospin coordinates are treated as collective coordinates. The classical kaon field is zero and the kaon fluctuations are treated as vibrational modes found by expanding the lagrangian to second order in the kaon fields. Solving for the kaon modes, it turns out that there exist b o u n d states in the background field of the SU (2) skyrmion. The Wess-Zumino ( W Z ) term plays an essential role in differentiating between the strangeness S = - 1 and S = + 1 solutions. Without this term in the lagrangian these solutions are degenerate. Including the WZ term, only the S = - 1 sector contains b o u n d states, the lowest one having an energy of about 150 MeV. This bound state mode for S = - 1 can be interpreted as a boson with isospin zero and intrinsic spin-parity j e = 1/2 +. These quantum numbers and the energy contribution per unit of strangeness correspond to that of a strange quark in baryons. Furthermore, the fine structure found in the semiclassical quantization of the skyrmion-kaon system perfectly matches the hyperfine splitting due to one gluon exchange in the quark models. In this letter we discuss the application of the bound state approach for the baryon number n sector o f the Skyrme model, mostly restricting ourselves to n =2. In order to find the kaon field fluctuations around the S U ( 2 ) soliton we follow CK, making the ansatz U=~U~,
(1)
where ~= ~fU. and UK = exp ( i 2 . . K J f ~ ) with 2a being the Gell-Mann SU (3) matrices and the index a running from 4 to 7. Identifying the standard (complex) kaon isodoublets K=~\K6_iKyj
=
K°
'
(2)
and substituting the above ansatz for U in the Skyrme lagrangian one finds after expanding to second order in the kaon fields [4,5] 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
449
V o l u m e 215, n u m b e r 3
50= ~F~Tr(c~U , ~c32
PHYSICS LETTERS B
22 D e c e m b e r 1988
1
* ~U=)+ 3Te2Tr[OuU~U=,O~Um~U=12+(D~K)*D~'K-m2K*K
(
1
)
+ IK*K T r ( ~ U ~ O ~ U ~ ) + 77-~z Tr[~U~U~, 8.U~U~] 2 eF~ 1
e 2 f ~ {2(D~K)*D.KTr(AUA~)+ I (D~K)*D~KTr(a.U~cq"U~)-6(D~,K)*[AU, A"]D. K} iNc
+ -~B~'[K*D~,K - ( D ~ K ) * K ] ,
(3)
where F~ = 2f~ and Au =
½(~ta~{- {a,,{*) ,
DuK= O~K+ ½(~tO#~-I-~a#~t)K.
(4) (5)
The last term in eq. ( 3 ) which is proportional to the number of colors Nc originates from the W Z action, which can be converted into an ordinary lagrangian term containing the baryon current 1
t p t t B , , - - - 24/r 2 E~,,p, Tr(0 v U=U=O U=U=Oa U~U=) .
(6)
For the background skyrmion we choose the variational ansatz discussed by Weigel et al. [ 6 ]. This ansatz as compared to the hedgehog ansatz is for n > 1 characterized by a different twist in the isovector field n rather than by a different boundary condition for the chiral angle F(r), i.e.
U=(r) =exp( i~.n/f~) ,
(7)
~z(r) = f ~ ( r ) F (
(8)
r) ,
//sin 0 cos n0~ ~.(r, 0, 0 ) = [ s i n 0 sin n 0 ~ , \ cosO /
(9)
and F ( o o ) = 0 , F ( 0 ) = m This ansatz is an approximation to the true axially symmetric solution found by Verbaarschot [ 7 ]. This approximation enables us to reduce the problem to a one-dimensional one. It leads to a soliton solution with a mass M2 = 2.14 Ml = 220 f J e , where f= and e are the parameters appearing in the Skyrme lagrangian [ 1,2 ], whereas the true solution has the m a s s m 2 ~ 1.92 M~. In the baryon n u m b e r one sector the invariance of the background field U,(r) under combined isospin and spatial rotations A = I + L enables a partial wave analysis of the kaon field in spinor harmonics YALAz"The energy eigenstates then are
K(r, t) = e x p ( - i o ) t ) k(r) YALA:(#) .
(10)
The lowest solution is found for L = 1, A = 1/2, in which case the spinor harmonics is YALA:(#) =T'#Z, where Z is a two-component isospinor. For n ¢ 1 the background field U~(r) is no more invariant under A. Therefore we restrict ourselves to the following consistent ansatz for stationary kaon states:
K(r, t) = v ' ~ . k = e x p ( - i o o t ) / 7 ( r ) v'~. Z,
( 11 )
where/7 is a function that depends only on r and Z is again a two-component spinor. For n = 1 this solution is precisely the lowest energy solution of CK. 450
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W i t h these Ans~itze in eqs. ( 7 ) - ( 9 ) a n d ( 11 ) for U. = U. [ F ] and K we obtain a spherically s y m m e t r i c lagrangian density d e p e n d i n g only on the radial coordinate. Choosing dimensionless variables with the energy scale set by eF. we find ~ = LfSk[ F ] +f(r)k*[c+i2(r)(k*!~-[c*k) - h ( r ) k ' * k ' - k * [ m ~
+ Veff(r ) ]k,
(12)
with Verr(r) = - ¼[ d ( r ) + ( n 2 + 1 )s(r) ] - 2 s ( r ) [n2s(r) + (n2+ 1 )d(r) ] 1 + 7~ [ n 2 + l + 2 n Z s ( r ) +
(n2+l)d(r)][c(r)-
112
6( n2s( r ) [ c ( r ) - l ] 2 + l ( n 2 + l ) d r d { [ c ( r ) - l ] f ' ( r ) s i n f ( r ) }
+~5
),
(13)
where f ( r ) = l + ( n 2 + l ) s ( r ) + d ( r ) , h(r)=l+(n2+l)s(r), 2 ( r ) = - ( e 2 N c / 2 7 r a r 2 ) s i n 2 ( F ) F ', s ( r ) = (sin F / r ) 2, d ( r ) = ( F ' ) 2 , a n d c ( r ) = sin 2 ( F / 2 ). A dot indicates a time derivative and a p r i m e a derivative with respect to r. F o r a given profile function F(r) one solves a second order differential equation f o r / ~ i n o r d e r to obtain stationary states o f the kaon field. The result for the energy eigenvalue e)~. o f the lowest kaon m o d e is shown in fig. 1 for the s k y r m i o n solutions o b t a i n e d by Adkins et al. [2] for n = 1 and by Weigel et al. [6] for n = 2. The p a r a m e t e r set for these solutions is e = 5.45 a n d F ~ = 129 MeV, which reproduces the nucleon and delta masses. The eigenvalues, shown as a function o f N~, d e m o n s t r a t e the i m p o r t a n c e o f the Skyrme term. W i t h the physical kaon mass, mK= 495 MeV, the energy o f the lowest energy solutions is a r o u n d 150-170 M e V for N~ = 3. This indicates that the basic energy difference per unit o f strangeness is quite similar for the cases n = 1 and n = 2, which is in agreement with quark models [ 8 ]. The fine structure o f the mass spectrum is o b t a i n e d by the semiclassical quantization procedure [2,6,4], where only isospin and spatial rotations are treated as zeromodes. We consider the time d e p e n d e n t spatial and isospin rotations R a n d A where the angular velocities with respect to the b o d y fixed axes are given by (R - ~k )~b = ~,~,&c,
(14)
15) The pion and the kaon fields transform according to
U . ~ R A U . A -~ ,
16)
K ~ RAK= ( RAv.~.A -l )Ak .
17)
0.5 n=l
. . . . . . .
-1'4- . . . . . . .
t~o.3 3 0.2
0.1
I
I
I
t
1
2
3
Z.
Nc
5
Fig. 1. Energy eigenvalue o) as a function of Nc for the lowest kaon and antikaon modes for n = 1 and n = 2. For N~= 0 the solutions are degenerate. For increasing values of No the S= + 1 solution is pushed up, the S= - 1 solution is pushed down in energy. The dashed line (M) indicates the kaon mass. 451
Volume 215, number 3
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Keeping in the expansion of the kaon field only the (two) eigenmodes in eq. ( 11 ) with coefficients a~ and a2 for the up and down spinor Z respectively, a straightforward exercise yields the lagrangian
L=Lsk +LK --T'/~+ 2dl [ (T~/~x+ T~/~v)- (Tx&x + Ty&v)0,,~ + T:(]~=-n&z) ] +2d2[n2(Txflx+Ty~,)-(T~&~+Ty&v)d~l + ½( 3 - n 2 ) ~(/~_.-n&~) ] ..}_10n [ (j~2 ...]._j~2) ...[_1(/,/2...[_3)(& 2 + & } ) - 2 (&xfl.,- + &y/~,,)a,,, + (/}: -n&z) 2]
-½a~[¼ (n2+3)(/}~.+,~2)+~(n4+3)(&2 +&{)-2(&x/~x+&,,g)&,l
+n2(g-n&z)
(18)
2] ,
where the explicit expressions for On= On [F] and J,, =An [F] are given in ref. [6], while d, = 8 n
f dr/~/~co ( -~fr2cos2(½f) - ~dr ( r 2 s i n F f ') ) ,
(19)
d2 = 8n f dr/~-*/~e)~ sin2F cos 2( ½F),
(20)
and T= a* ~'%/2, summed over the modes. The kaon modes are normalized according to 8n f dr/~*/~(f~o+ 2) = 1. The quantities & and/) are the angular velocities with respect to the body fixed axes. The spin components j b f and the isospin components I b~ relative to body fixed axes are given by jbf=0L/O&;,
(21)
i br=0L/0fl,.
The symmetry of the skyrmion solution gives rise to constraints. For n = 1 these are
Jbf=--(Ibr+T;)
or J + I = - T ,
(22)
while for n ~ 1 only one constraint due to the axial symmetry of the classical solution is left, j b__ f =
--n(Ibf + Tz) .
(23)
The quantization of the rotational energy leads to 1
c
2
1-c
It~o,= ~ (I+cT)2= f ~ S + _~_~_i2+
c ( c - 1) 2n T:,
(24)
n=l,
1 1 _ __ (d2_j~) + ~ ( F _ I 2) + ~c22 ( T 2 _ T ~ ) + ~ 31 (I:+c~ Tz)2+ ~c2 7 (I+ T_ +I_ T+),
n#l,
2-Ql
(25) where the spin and isospin components are with respect to body fixed axes and g2~ = 1 (n2+ 3)On - ¼(r/4+ 3)Jn,
(26)
g22 = O ~ - ¼(n2+3)An,
(27)
~¢~3= On -- g/2An ,
(28)
c, = 1 - 2 [ d l + l ( 3 - - n 2 ) d 2 ] ,
(29)
c2 = 1 - 2 ( d , +n2d2).
(30)
For n = 1 the O's converge into one expression g2 and Cl and c2 converge into c. The results for c, c~ and c2 are shown in fig. 2. The above expression for the rotational energy is very general and includes as limiting cases the SU(2) Skyrme models of ref. [2] ( n = l , T = 0 ) and ref. [6] ( n ¢ 1, T = 0 ) and the result of ref. [5] ( n = l , T~ 0). The above system is similar to the particle-rotor system known in nuclear physics [ 9 ]. 452
22 December 1988
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Volume 215, number 3 1.0
0.5
0.0
I
Fig. 2. The parameters c, cj and c2 [see eqs. (25), ( 29 ) and (30) ] as a function of No, We now turn to the results for T # 0. For the n = 1 case the mass formula, eq. (24), is a special example of the general mass formula, M = mo + m~ Y + m z J ( J + 1 ) + m3 [ I ( I + 1 ) - ~ y2 ] + m4 y 2 ,
(31 )
which is based on SU (6) s y m m e t r y breaking [ 8 ] and which, up to the term oc y2, dates back to the early sixties [ 10 ]. The equivalence can easily be seen by rewriting the spin of the strange mesons in terms of strangeness or hypercharge, T = ISI/2= t Y - 1 I/2. The mass formula in the bound state approach to the Skyrme model, however, is slightly more restrictive involving only four quantities, namely Md, ogr, c, and I2. It turns out that considering these four quantities as free parameters, an excellent fit to the baryon spectrum can be obtained, yielding Mc~=0.867GeV,
wr=215MeV,
1/2£2=99.7MeV,
c=0.661.
The result for c in this fit is in good agreement with the calculated value of 0.62 found for No= 3 (see fig. 2). The result for co~cis farther o f f f r o m the calculated result (see fig. 1 ). The eigenfunctions of Hrot m a y be constructed as the product states of the rotation matrices for isospin and spin and the kaon eigenstates, ( a,, fli, r, t l L L; J, J:; Ib~f~= K - T~, j_bf= - n K , Tz ) =
[ (2•+ 1 ) ( 2 J + 1 ) 11/2 8~ 2
DSj:_,l~( a)D~:t~_T:(fl)kT:(r, t) . (32)
For T = 0 the states I L I z ; J , J : ; I The hamiltonian is
bf: =t%" jbf: = - n K ,
I ( I + 1) 2(1 //rot= J ( J + l ) + _ _ +K 212~ 25"22 2~s
1 2f2z
T:=0)
are the eigenstates of Hro, as discussed in ref. [6].
n2 ) 2/2t "
(33)
For T # 0 the eigenstates are obtained after diagonalization of Hrot. For strangeness S = - 1 the intrinsic spin of the strange meson is T = 1/2 and the hamiltonian in general is a 2 × 2 matrix. Choosing the basis states I1, I ~ ", J , J : , I.~ br = K +-- I / 2 , Jzbf = -- nK, T: = ___1 / 2 ) , the hamiltonian is given by [K--½(1-Cl)] Hrot = J ( J + 1 ) I(I+ 1 ) 2~----~ + 2(22
n2K 2 c2 2a~ + -~2 +
(K-½) 2
2~Q3 2g22 c2x/I(I+l)-K2+~ 2922
c2x/I(I+I)-K2+~ 2g22 [K+½(1-c,)] 2 2-('23
(K+½) 2 2922
(34) 453
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F o r strangeness S = - 2 the i n t r i n s i c spin o f the strange m e s o n s m u s t be T = l, since the k a o n s are bosons, a n d the h a m i l t o n i a n is a 3 X 3 matrix. W i t h t h e basis states [ I , L ; J , J ~ ; I:bf =K, j b r = - n K , T ~ = O ) , and 1I, I~; J, J:; I bf = K-T- 1, jbr = _ nK, T~ = + 1 ) , o r d e r e d a c c o r d i n g to i n c r e a s i n g T: values, we o b t a i n
J(Jq- 1 ) H~ot- - 2121
+
I(I+ 1)
rt2K 2
-
- -
-
2122
(K--1+C,) 2 2-Q3 +
2£21
(K-l)
2
c2 + - -
2g-22
c2x/I(l+ 1)-K(K-
2-Q2
c 2 x / I ( I + 1) - K ( K -
1)
x/2 a2
1)
0
x / 2 -(22
K: 2a3
K~
c2
c2x/I(l+l)-K(K+l) 0
c 2 ~ / I ( I + 1) - K ( K +
2a~ + 2a~
x f 2 £22
1)
(35)
xf2 a2
(K+l-Cl)
2
(K+I) 2
2923
2 ~"22
In all these cases the v a l u e s o f I a n d J are l i m i t e d by I >/[ I bf [ a n d J>~ ]j b f ]. F o r n = 2 the parity o f the states is g i v e n by P = ( - ) ~, w h i c h can be o b t a i n e d in a w a y a n a l o g o u s to ref. [ 6 ], a n d K m u s t be integer in o r d e r to f i n d p a r i t y eigenstates. T h e resulting lowest states for b a r y o n n u m b e r 2 in the S = - 1 a n d S = - 2 sector for Arc= 3 using the s a m e v a l u e s for e a n d F , as b e f o r e are g i v e n in table 1. T h e r e l e v a n t v a l u e s for the q u a n t i t i e s in eqs. ( 2 6 ) - ( 3 0 ) are 1 / 2 ( 2 1 - - 3 8 M e V , 1 / 2 £ 2 2 = 6 0 M e V , 1 / 2 g ~ 3 = 7 7 MeV, Cl = 0 . 6 8 a n d c 2 = 0 . 5 7 . T h e results in the n = 2 sector s h o w t h a t in each o f the c h a n n e l s there are a n u m b e r o f states for w h i c h //rot < Hrot (B l) + Hrot (B2), w h e r e B~ a n d B2 are the lowest energy b a r y o n s to w h i c h the s y s t e m can couple. Taking i n t o a c c o u n t the d i f f e r e n c e b e t w e e n t h e classical energies o f the soliton, M n = 2 - 2 M , = 1~ 0.14Mn= 1~ 120 M e V a n d the d i f f e r e n c e b e t w e e n the k a o n m o d e energies o 0 ~ ( B = 2 ) - ~ % ( B = 1 ) ~ 10 M e V , no b o u n d states
Table 1 Energy spectrum for S= 0, - l and - 2. Given are the eigenvalues of H~ot in MeV for the n = 2 states or between brackets the sum of the eigenvalues of Hrot for the lowest baryon-baryon channels. Flavor assignments, although not unique, have been indicated in order to facilitate comparison with the spectral content in the collective quantization procedure. States that do not appear in the quark model are indicated with (e) S=0
S=-I
S=-2
f
(1, JP)
(Hro,)
f
(I, je)
(Hro,)
10"
(0, 0 + )
10"
(0, 1 + )
27
( 1, 0 + ) I=0, 1 (1, 1+) (1,2-) (0, 2 + )
0 (e) 76 120 147] 196 (e) 212 227 (e)
10" 27 10"
(1/2,0 + ) (1/2,0 + ) (1/2,1 + ) I= 1/2 (1/2,1 + ) (3/2,0 + •=3/2 (1/2,2(1/2,2 + (3/2,1 + (3/2,2
76 83 90] 152 (~) 160 165] 169 234 (e) 235 (e) 245
[NN 27 27 10"
[AN 27 27
[ZN 27 10" 27 27
454
8 I~)
f
(i, je)
(Hro,)
10"
( 1, 0 + )
[AA
I=0 (0, 0 + ) ( 1, 0 + ) (1, 1+ ) 1= 1 (0, 1+) (0, 2- ) (1, 1+) (1, 2-) 1=2 (2, 0 + ) ( 1, 0 + ) (1, 2 + ) (0, 2 + ) (2, 1+ )
20 (~) 33] 39 88 96 108] 115 (e) 130 163 (e) 182 183] 192 227 (~) 247 (~) 266 267 (¢)
27 27 10" [XA 27 27 27 27 [£'X 27 35 10" 27 27
Volume 215, number 3
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remain in the S = - 1 and S = - 2 sectors. As was already pointed out, however, the classical solution in the baryon number 2 sector can be improved upon [ 7 ], in which case a number of states actually could become bound. In order to classify and interpret the above n = 2 states it is illuminating to compare with the spectra of the corresponding SU (3) collective quantization and o f the quark model. In the collective quantization scheme the presence of the W e s s - Z u m i n o term leads to a constraint on the allowed flavor representations in the form of a constraint on the right (or body fixed) fixed hypercharge, YR=Ncn/3 [ 11 ]. For n = 1 and N c = 3 , YR= 1 and therefore the allowed flavor multiplets f with the lowest energy are 8 and 10 [we indicate SU (3) irreducible representations by their dimensions ], which because of YR = 1 have J = 1/ 2 and J = 3 / 2, respectively. In the n = 2 sector, the constraint YR = 2 yields flavor multiplets 10", 27, 35 and 28 as the lowest energy states. Since the classical solution has only axial symmetry, however, this leads to no further restriction on the spin than it being an integer. Although it is not possible to make unique flavor assignments in the bound state approach to dibaryons, in essence the same spectrum appears. The lowest states in table 1 can be accounted for as the members o f (10", 0 + ), (27, 0 + ), (10", 1 + ), and (27, 1 +1 ) flavor-spin multiplets. This Skyrme model spectrum differs from the quark model spectrum in two aspects. On the one hand it contains multiplets that are forbidden in the quark model. The (10", 0 + ) and (27, 1 + ) are such examples, that correspond to the (L je) = (0, 0 + ) and ( 1, 1 + ) states in the nonstrange case [6]. In the S = - 2 sector, we find for instance two ( 1, 0 + ) states, while there is only one such state in quark models. On the other hand the quark model contains different multiplets in the n = 2 sector, which do not necessarily contain Y= 2 states. Such examples are the 1, 8 and the 10. A state of special interest in the S = - 2 sector is the flavor singlet H-dibaryon with (L JP) = (0, 0 + ) quantum numbers. In quark models it is appreciably lower than other states in the spectrum [ 12 ]. We also found a state with the q u a n t u m numbers (0, 0 + ) of the H-dibaryon, but it is not the lowest one in the S = - 2 sector. Furthermore, the similarity of the spectra in the bound state approach and in the collective quantization suggests that this (0, 0 + ) state belongs to the (27, 0 + ) multiplet. In the S U ( 3 ) collective quantization scheme the low-lying H-dibaryon is not based on the SU (2) embedding but on the SO ( 3 ) embedding in SU (3), i.e. on a very different classical solution [ 11,13,14 ]. Obviously, we should not find the corresponding q u a n t u m states, (1, 0 + ), (8, 1 + ), (8, 2 + ) (10, 1 + ) and (10, 3 + ), with our embedding. In this paper we have extended the b o u n d state approach to strangeness in the Skyrme model to the baryon n u m b e r two sector using the classical solution of ref. [ 6 ] as an approximation to the classical ground state in that sector. The bound state approach gives a very satisfactory qualitative description of the baryon number one sector. For the qualitative agreement one does not need to introduce heavier mesons explicitly, although they might improve the quantitative result [ 15 ]. In the baryon number two sector one finds a spectrum of states similar as the one obtained in the collective quantization scheme starting with a classical SU (2) solution embedded in SU (3). This does not yield a low-lying bound state in the S = - 2 sector corresponding to the H-dibaryon. The authors would like to acknowledge discussions with M. Rho (Saclay). This work is in part (J.K.) supported by the Deutsche Forschungsgemeinschaft ( D F G ) and in part (P.J.M.) by the Foundation for Fundamental Research ( F O M ) and the Netherlands Organization for the Advancement of Scientific Research ( N W O ) .
References
[ 1] T.H.R. Skyrme, Proc. R. Soc. A 260 ( 161 ) 127; Nucl. Phys. 31 (1962) 556. [2] G.S. Adkins, C.R. Nappi and E. Winen, Nucl. Phys. B 228 (1983) 552. [ 3 ] P.O. Mazur, M.A. Nowak and M. Praszatowicz, Phys. Lett. B 147 ( 1984) 137; M. Praszalowicz, Phys. Lett. B 158 ( 1985 ) 264; M. Chemtob, Nucl. Phys. B 256 (1985 ) 600. [4] C.G. Callan and I. Klebanov, Nucl. Phys. B 262 (1985) 365. 455
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[5] C.G. Callan, K. Hornborstel and I. Klebanov, Phys. Lett. B 202 (1988) 269. [ 6 ] H. Weigel, B. Schwesinger and G. Holzwarth, Phys. Lett. B 168 ( 1986 ) 321. [7] J.J.M. Verbaarschot, Phys. Lett. B 195 (1987) 235. [8] A.T. Aerts, P.J. Mulders and J.J. de Swart, Phys. Rev. D 17 (1978) 260. [ 9 ] A. Bohr and B. Mottelson, Nuclear structure, Vol. II (Benjamin, Reading, MA, 1975 ). [ 10 ] F. Giirsey and L. Radicati, Phys. Rev. Lett. 13 (1964) 173; A. Pais, Phys. Rev. Lett. 13 (1964) 173; B. Sakita, Phys. Rev. 136 (1964) B1756; T.K. Kuo and T. Yao, Phys. Rev. Lett. 13 (1964) 415; M.A.B. Bdg and V. Singh, Phys. Rev. Lett. 13 (1964) 418. [ 11 ] A.P. Balachandran, F. Lizzi, V.G.J. Rodgers and A. Stern, Nucl. Phys. B 256 ( 1985 ) 525. [12] R.L. Jaffe, Phys. Rev. Lett. 38 (1977) 195. [ 13 ] S.A. Yost and C.R. Nappi, Phys. Rev. D 32 ( 1985 ) 816. [ 14] J. Kunz and D. Masak, Phys. Lett. B 191 (1987) 174. [ 15 ] N. Scoccola, H. Nadeau, M. Nowak and M. Rho, Phys. Lett. B 201 ( 1988 ) 425.
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